# A Polynomial Degree Bound on Defining Equations of Non-rigid Matrices and Small Linear Circuits

We show that there is a defining equation of degree at most 𝗉𝗈𝗅𝗒(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field 𝔽, there is a non-zero n^2-variate polynomial P ∈𝔽(x_1, 1, …, x_n, n) of degree at most 𝗉𝗈𝗅𝗒(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and sparsity at most n^2/100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [GHIL16] and improves the best upper bound known for this problem down from (n^2) [KLPS14, GHIL16] to 𝗉𝗈𝗅𝗒(n). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n^2/200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [SV15] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof.

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